C++ for Scientists - Technische Universität Dresden

2.11. REAL-WORLD EXAMPLE: MATRIX INVERSION 57
for (unsigned k= 0; k < n; ++k) {
}
In each iteration we need the k-th unit vector.
dense vector e k(n);
for (unsigned i= 0; i < n; ++i)
if (i == k)
e k[i]= 1.0;
else
e k[i]= 0.0;
The triangular solver returns a column vector. We could assign the entries of this vector directly
to entries of the target matrix:
for (unsigned i= 0; i < n; ++i)
Inv[i][k]= upper trisolve(A, e k)[i];
This is nicely short but we would compute upper trisolve n times! Although we said that performance
is not our primary goal at this point, the raise of overall complexity from order 3 to 4 is
too much waste of resources. Therefore, we better store the vector and copy the entries from
there.
dense vector res k(n);
res k= upper trisolve(A, e k);
for (unsigned i= 0; i < n; ++i)
Inv[i][k]= res k[i];
Return our temporary matrix finishes the function that we now give in its complete form.
dense2D inverse upper(dense2D const& A)
{
const unsigned n= num rows(A);
assert(num cols(A) == n); // Matrix must be square
}
dense2D Inv(n, n);
for (unsigned k= 0; k < n; ++k) {
dense vector e k(n);
for (unsigned i= 0; i < n; ++i)
if (i == k)
e k[i]= 1.0;
else
e k[i]= 0.0;
dense vector res k(n);
res k= upper trisolve(A, e k);
for (unsigned i= 0; i < n; ++i)
Inv[i][k]= res k[i];
}
return Inv;